Hardy spaces and Laguerre expansions on the dual of the Heisenberg group

Sia D il dominio di Siegel e H_n il gruppo di Heisenberg. Si considera il sistema ortogonale ottenuto dai monomi normalizzati delta palla unitaria di ⁿ⁺¹ tramite la trasformata di Cayley. La trasformata di Fourier di tali funzioni ristrette ad H_n viene esplicitamente calcolata. Rispetto alla base di Hermite di L²( ⁿ ), si ottengono operatori di rango uno espressi in termini di funzioni di Laguerre.     

Sobolev spaces on Lie groups: Embedding theorems and algebra properties

Let G be a noncompact connected Lie group, denote with ρ a right Haar measure and choose a family of linearly independent left-invariant vector fields X on G satisfying Hörmander's condition. Let χ be a positive character of G and consider the measure ${\mu }_{\chi }$ whose density with respect to ρ is χ. In this paper, we introduce Sobolev spaces $L_{\alpha }^p({\mu }_{\chi })$ adapted to X and ${\mu }_{\chi }$ ($1<p<\infty$, $\alpha \ge 0$) and study embedding theorems and algebra properties of these spaces. As an application, we prove local well-posedness and regularity results of solutions of some nonlinear heat and Schrödinger equations on the group.     

Absolute and relative cut-off in adaptive approximation by wavelets

Given the wavelet expansion of a function v, a non-linear adaptive approximation of v is obtained by neglecting those coefficients whose size drops below a certain threshold. We propose several ways to define the threshold: all are based on the characterization of the local regularity of v (in a Sobolev or Besov scale) in terms of summability of properly defined subsets of its coefficients. A-priori estimates of the approximation error are derived. For the Haar system the asymptotic behavior of both the approximation error and the number of survived coefficients is thoroughly investigated for a class of functions having Hölder-type singularities.     

Stereochemistry of nucleophilic substitution reactions depending upon substituent: evidence for electrostatic stabilization of pseudoaxial conformers of oxocarbenium ions by heteroatom substituents

Lewis acid-mediated nucleophilic substitution reactions of substituted tetrahydropyran acetates reveal that the conformational preferences of six-membered-ring cations depend significantly upon the electronic nature of the substituent. Nucleophilic substitutions of C-3 and C-4 alkyl-substituted tetrahydropyran acetates proceeded via pseudoequatorially substituted oxocarbenium ions, as would be expected by consideration of steric effects. Substitutions of C-3 and C-4 alkoxy-substituted tetrahydropyran acetates, however, proceeded via pseudoaxially oriented oxocarbenium ions. The unusual selectivities controlled by the alkoxy groups were demonstrated for a range of other heteroatom substituents, including nitrogen, fluorine, chlorine, and bromine. It is believed that the pseudoaxial conformation is preferred in the ground state of the cation because of an electrostatic attraction between the cationic carbon center of the oxocarbenium ion and the heteroatom substituent. This analysis is supported by the observation that selectivity diminishes down the halogen series, which is inconsistent with electron donation as might be expected during anchimeric assistance. The C-2 heteroatom-substituted systems gave moderately high 1,2-cis selectivity, while small alkyl substituents showed no selectivity. Only in the case of the tert-butyl group at C-2 was high 1,2-trans selectivity observed. These studies reinforce the idea that ground-state conformational effects need to be considered along with steric approach considerations.     

Analytic Features of Reproducing Groups for the Metaplectic Representation

We introduce the notion of admissible subgroup of relative to the (extended) metaplectic representation via the Wigner distribution. Under mild additional assumptions, it is shown to be equivalent to the fact that the identity holds (weakly) for all We use this equivalence to exhibit classes of admissible subgroups of We also establish some connections with wavelet theory, i.e., with curvelet and contourlet frames.     

Dimensional upper bounds for admissible subgroups for the metaplectic representation

We prove dimensional upper bounds for admissible Lie subgroups H of G = ?^d ? Sp (d, ?), d ≥ 2. The notion of admissibility captures natural geometric phenomena of the phase space and it is a sufficient condition for a subgroup to be reproducing. It is expressed in terms of absolutely convergent integrals of Wigner distributions, translated by the affine action of the subgroup. We show that dim H ≤ d² + 2d, whereas if H ? Sp (d, R), then dim H ≤ d² + 1. Both bounds are shown to be optimal (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multilevel decompositions of functional spaces

A unified abstract framework for the multilevel decomposition of both Banach and quasi-Banach spaces is presented. The characterization of intermediate spaces and their duals is derived from general Bernstein and Jackson inequalities. Applications to compactly supported biorthogonal wavelet decompositions of families of Besov spaces are also given. The first author was partially supported by grants from MURST (40% Analisi Numerica) and ASI (Contract ASI-92-RS-89), whereas the second author was partially supported by grants from MURST (40% Analisi Funzionale) and CNR (Progetto Strategico “Applicazioni della Matematica per la Tecnologia e la Società”).     

Dipeptidyl Peptidase 3 Activity as a Promising Biomarker of Bone Fragility in Postmenopausal Women

The dipeptidyl peptidase 3 (Dpp3) is a ubiquitous zinc-dependent aminopeptidase, participating in the activation or degradation of signaling peptides and in the Keap1–Nrf2 antioxidant pathway. The absence of Dpp3 in the Dpp3 knockout mouse model causes increased osteoclast activity, altered osteogenic function, sustained oxidative stress in the bone tissue, and bone loss. We aimed to assess the association of Dpp3 activity with bone fragility in postmenopausal osteoporosis and the impact of denosumab on enzymatic activity. We conducted a two-phase study including 69 postmenopausal women with severe osteoporosis and 36 postmenopausal women without osteometabolic conditions, as controls (cross-sectional phase). Subjects with severe osteoporosis were assessed at baseline and 14 days after the first denosumab administration (prospective phase). The results showed significant reduction in serum Dpp3 activity (expressed as nmoles of formed product/mg proteins/min) in patients vs. controls (0.791 ± 0.232 vs. 1.195 ± 0.338; p < 0.001), and significant association with bone mass at the femoral neck (r = 0.28, p = 0.02) in patients prior to treatment. We found a negative correlation between C-terminal telopeptide (CTX) or N-terminal pro-peptide of type 1 procollagen (P1NP) levels and Dpp3 activity (respectively, r = −0.29, p = 0.012; and r = −0.2572, p = 0.033). Dpp3 activity did not change after denosumab injection. Our findings support a critical role played by Dpp3 in bone homeostasis as a potential bone protective factor. Additional clinical studies in larger cohorts might explore the implementation of Dpp3 assessment as a biomarker of bone health status.     

Riesz Transform for a Flow Laplacian on Homogeneous Trees

Journal of Fourier Analysis and Applications - We introduce and study the notion of weak semi-greedy systems—which is inspired in the concepts of semi-greedy and branch semi-greedy systems...